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Self-dual code over \(R\)

Description

An additive linear code \(C\) over a ring \(R\) that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to some inner product.

Cousins

  • Unimodular lattice— There are parallels between self-dual codes over \(\mathbb{Z}_{2k}\) and even unimodular lattices [1,2].
  • Niemeier lattice— Extremal Type II self-dual codes of length 24 over \(\mathbb{Z}_6\) yield Niemeier lattices [3].

Member of code lists

Primary Hierarchy

Classical Domain
Ring Kingdom
Ring codeECC
\(R\)-linear codeECC
Dual code over \(R\)
Parents
Dual code over \(R\)
Self-dual code over \(R\)
Children
Self-dual additive codeSelf-dual linear
Harada-Kitazume code
Harada-Kitazume codes are extremal Type II self-dual codes over \(\mathbb{Z}_4\) [4].
Octacode
The octacode is self-dual over \(\mathbb{Z}_4\).
Pseudo Golay code
Pseudo Golay codes are Type II self-dual codes over \(\mathbb{Z}_4\) [5; Thm. 9].

References

[1]
A. Bonnecaze, P. Sole, C. Bachoc, and B. Mourrain, “Type II codes over Z/sub 4/”, IEEE Transactions on Information Theory 43, 969 (1997) DOI
[2]
E. Bannai, S. T. Dougherty, M. Harada, and M. Oura, “Type II codes, even unimodular lattices, and invariant rings”, IEEE Transactions on Information Theory 45, 1194 (1999) DOI
[3]
M. Harada and M. Kitazume, “Z6-Code Constructions of the Leech Lattice and the Niemeier Lattices”, European Journal of Combinatorics 23, 573 (2002) DOI
[4]
M. Harada and M. Kitazume, “Z4-Code Constructions for the Niemeier Lattices and their Embeddings in the Leech Lattice”, European Journal of Combinatorics 21, 473 (2000) DOI
[5]
E. Rains, “Optimal self-dual codes over Z4”, Discrete Mathematics 203, 215 (1999) DOI
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Zoo Code ID: self_dual_over_rings

Cite as:
“Self-dual code over \(R\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/self_dual_over_rings
BibTeX:
@incollection{eczoo_self_dual_over_rings, title={Self-dual code over \(R\)}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/self_dual_over_rings} }
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Permanent link:
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Cite as:

“Self-dual code over \(R\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/self_dual_over_rings

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/rings/dual/self_dual_over_rings.yml.